Financial Analysis puzzles, solved step by step

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  1. 038You have a 3-litre bottle, a 4-litre bottle and a tap. Neither bottle has markings. How do you measure exactly 2 litres?Logic and counting brainteasersWarm upNomuraNew York · 2026

    Try it first

    What is the fewest number of fills and pours that leaves exactly 2 litres?

    Show the worked solution

    Fill the 3-litre bottle, pour it into the 4, fill the 3 again and pour until the 4 is full: 2 litres are left in the 3-litre bottle. The 4 already holds 3 litres, so it takes only 1 more. That is four moves. A longer route fills the 4 first and ends with 2 litres in the 4-litre bottle after six moves.

    How do you avoid getting lost halfway through?

    Think of following a recipe that says pour half the milk into the other pan: if you do not write down how much is in each pan, two steps later you have no idea. Track the pair of levels after every single move, written as (3-litre, 4-litre), and the puzzle becomes a short walk through a few states instead of a juggling act. There are only three kinds of move: fill a bottle from the tap, empty a bottle, or pour one into the other until the source is empty or the target is full.

    Write down both bottles after every move; 2 litres appears on move four3 L04 L0Start3 L34 L01. Fill the 33 L04 L32. Pour 3 into 43 L34 L33. Fill the 33 L24 L44. Top up the 4Longer route, six moves: fill 4, pour into 3 (1 left), empty 3, pour the 1 across,fill 4, top up the 3 (needs 2): 2 litres left in the 4 litre bottle.
    Filling the 3-litre bottle, pouring it into the 4, filling the 3 again and topping up the 4 leaves exactly 2 litres in the 3-litre bottle after four moves, because the 4-litre bottle had room for only 1 more litre.

    Say the states out loud as you go: (0, 0), (3, 0), (0, 3), (3, 3), (2, 4). The interviewer can check each one in a second, and if you slip, you can see where.

    Which amounts can these two bottles measure at all?

    Every pour moves water in steps of 3 and 4, so the amounts you can isolate are the combinations of 3 and 4 with whole numbers in front: 4 minus 3 is 1, 3 times 2 minus 4 is 2, and so on. Because 3 and 4 share no common factor bigger than 1, these bottles can measure every whole number of litres from 1 to 4, and up to 7 if you count both bottles together. With a 4-litre and a 6-litre bottle you could never get an odd number, since every amount would be a multiple of 2. Saying that rule turns a party trick into reasoning.

    Where candidates lose it

    Candidates start pouring in their head without writing states and lose track after three moves, then restart. It looks like panic even when it is not. Say the pair of numbers after each move and the interviewer follows you.

    The second loss is stopping at a long route without checking for a short one. Mentioning that you found 2 litres in four moves, and that a six-move route also works, shows you looked for the efficient answer.

    What the interviewer asks next

    • With the same bottles, how do you measure exactly 1 litre?
    • With a 6-litre and a 9-litre bottle, can you measure 4 litres?
    • How would you prove that a given amount cannot be measured?

    Asked at Nomura, Equity Capital Markets, New York, 2026 (Wall Street Oasis): How much water can you fill using 1 3liter and 1 4liter bottle using each other?